 Hello everyone, thank you very much for coming. So yes, this can be a mathematical view of voting systems. So the great thing about maths is that it gives you this really precise language that we can express things in. So we can write statements that have no ambiguity, which is different from normal speech. So in normal speech you might say the voting system isn't fair, but that could have various different meanings. But we can actually apply specific, precise, unambiguous maths language to voting systems, and we can make definitions, proof theorems, provide examples and counter examples, and do all of that to show stuff about voting systems. So an electorate is defined as a set of voters, and voters have transitive preferences. So transitive means if a voter prefers x to y and prefers y to z, then they must prefer x to z. But even though all the people in the electorate have transitive preferences, it doesn't mean that the electorate overall will have transitive preferences. So the table there gives you an example of one where there aren't transitive preferences. Two people prefer x to y, two people prefer y to z, and two people prefer z to x. So there's no resolution here of making a transitive preference system that pleases everyone. So a voting system is just defined as an algorithm, so it takes in a set of n, so suppose it's n voters, it takes in their preferences, does some computation and outputs ranking of the candidates. So formally, if you have a set of n possible rankings of candidates, the algorithm is a function that outputs just a single ranking of candidates from them. So for example, first plus the post is the simplest one. Any preferences except for first preferences are just ignored, so the voters are only allowed to give their first preference, and whichever of the first preferences is most popular just wins straight away. So that's an example of probably the simplest voting system. So if there are exactly two candidates in an election, there is good news provided by May's theorem. So suppose we want a voting system, there's two candidates, and we want every voter to have the same influence. So on the screen I have a way of expressing this in precise mathematical language. So each voter has the same influence. The winning conditions are identical for both candidates. So for example, if one candidate needed 60% of the vote to win, that wouldn't be at the same condition for both candidates. And if C2 is not more popular than C1, if candidate 2 is not more popular than candidate 1, and then candidate 1 becomes more popular, candidate 1 must still win. So these are three fairly straightforward, fairly minimal requirements for a voting system. And May's theorem says first plus the post where everyone just votes which is their favourite or abstains if they're indifferent between them, and then you go with whichever is the most popular. That is the only system that satisfies all these criteria. So that's the good news. And so a brief side look at changing the status quo when there are kind of two candidates, but also kind of aren't two candidates. So suppose there's multiple alternatives to the status quo. So for example, if you wanted to say, should we change the national anthem or something, you might say, well, maybe, but it depends what we're going to change it to. So you might say, I'm not too unhappy about changing it, but there would have to be a good alternative anthem. So the obvious problem with just having a should we change it or not problem is that, let's say, just under a majority of people could favour staying with the status quo, and the status quo could be any alternative in a one-on-one competition, but it could still lose if we just have a simple should we change or not. And so after we've moved to the alternative and then choose which of the alternatives we want, there could be a majority who are saying, actually, I would have rather we just stayed with the original one. So the solution is similar to New Zealand's referendum on changing their flag in that they had a vote first on what the alternative flag should be and then put it up against the current flag in a one-on-one vote so that a majority would definitely be happy with the decision. OK, so we have the good news with May's theorem. Let's look at the bad news if there are more than two candidates. So we want every voter, if every single voter prefers candidate X to candidate Y, the voting system must say X is better than Y. Thus if every single voter agrees, then we must have that in the output. And now the second condition, independence of irrelevant alternatives, if X is ranked, if the voting system says X is better than Y, and then some other candidate Z, that's not X or Y, some other candidate becomes more popular, then X is better than Y can't change. So suppose you're choosing a candidate for a job and you say, yeah, well, I think the first candidate was definitely better than the second candidate, we agreed on that, and then a third candidate comes along who doesn't tell you anything about candidates one and two, and you say, actually now I've seen candidate three, I think candidate two is better than candidate one. This extra information that doesn't tell you anything about the first two candidates shouldn't affect your opinion of them, and there should be no dictator, so there shouldn't just be a single voter who says, oh, if I'm going, I vote, whichever way I vote, that's the way the election will go, so there's no prespecified person who, they're the only vote that counts. So there's no system that satisfies these three conditions, and the most commonly unsatisfied condition is the second one, independence of irrelevant alternatives, so let's have a look at that in first-past-the-post. So suppose you've got these preferences, so there's 110 people who like X, Z, Y, 102 like Y, X, Z, and eight people like Z, X, Y. So suppose nine people change their preference from the first one to the third one, then Y would win. Y is still as unlike as before, people still say anyone would be better than Y, I prefer X to Y, I prefer Z to Y, anyone but Y, but because some people say actually, yeah, maybe Z is the better of X and Z, now I win. So this violents independence of irrelevant alternatives, and also violates the majority loser criterion, so you can elect unpopular candidates, you can elect candidates who say, yeah, a majority of people say anyone but this candidate and they can still win. So let's have a look at the alternative vote. So just a quick definition of it, the voters rank their candidates rather than just putting a first preference, you count up the first preferences, if someone has a majority then they immediately win, otherwise whoever has the fewest votes gets knocked out and their second preferences turn into first preferences and then you just repeat it until someone has a majority and they win. So this one, wait. Oh yeah, sorry, this one. Okay. So the problem with first pass the post is that you can have these clones, so if two candidates are ranked next to each other and one person ZR, then they harm each other because they split the vote. So if you want to win a first pass the post election, just clone your opponent. Just make a candidate that's really similar to them. So AV is clone proof. If you have two candidates who are ranked, who are always ranked next to each other, people say, yeah, these two candidates are roughly the same. If one of them didn't run, it wouldn't make any difference to the other candidates. So that's guaranteed mathematically. It also satisfies majority loser criterion. So if, as one candidate, more than half the people say anyone but that candidate then they can't win. So is AV just a better system? No. So suppose that we have an election and X gets some ranking, let's say X wins, and then the electorate change and they say actually we like X even more now and we have the second election because let's say X wins the first election, then X serves their term and then they're even more popular next time around. That shouldn't hurt X. X should just win again, right? But AV isn't monotone. That's a monotone condition. AV isn't monotone. If a candidate becomes more popular, it can actually harm them, which is a particular vulnerability. AV has this problem. First pass the post doesn't. So let's give an example. So in the first vote, we have W has nine votes, X has six, Y has six and Z has five. So Z is knocked out. Then W has nine votes, X has 11 and Y has six. So Y goes out, Y's votes get sent to X. And so in the final result, W has nine and X has 17. So X would win that vote. Then suppose the voters change to voters prime with the apostrophe. So now in the first round, Y is the least popular candidate rather than Z. So Y is eliminated first. Y's second votes go to Z. So now it's W has nine, X has eight and Z has nine. So X gets knocked out in the second round. So now in there votes transferred to Z. So a Z wins with a majority now. So the same majority is X had before. So X lost by becoming more popular in this case. And it also doesn't satisfy the participation criterion. So you can be in the situation where you get a better outcome for yourself if you don't vote at all. So let's say there's kind of three, there's three, Tennessee is choosing a new state capital. We want to be near to the capital as possible and live in the capital if they can. So there's three kind of cities on one side and Memphis, which is right in the bottom corner, that is the most populous. So we've got Memphis's majority loser. The three other cities would rather have anyone but Memphis because they're all closer to each other. So straight away, well to me it seems obvious that Nashville is the kind of compromise winner. Should be the compromise winner here. Because it's kind of in the middle and it has a fair amount of support on its own anyway. But the problem is that Memphis voters, because they're popular but not quite unpopular enough to get knocked out, their votes never get reassigned. They're on 42%. So they're not going to be eliminated. So what happens is Chattanooga voters get knocked out, their votes go to Knoxville, Nashville voters, Nashville is then the least popular, their votes get knocked out, they go to Knoxville. So Knoxville wins, Knoxville wins here even though it's clearly kind of the extreme candidate at the right hand side away from everything else. So what could we have done differently? Well if Memphis voters, if the figure had been 25% of original voters rather than 42%, if that was what Memphis, if people had stopped, if fewer people had voted from Memphis, they would have been knocked out and their votes would have gone to Nashville and Nashville would have won. So people from Memphis who voted for Memphis harmed their own interests because a few of them had voted a better candidate for them would have won. So basically with two options you do have a very good voting system, just choose your favourite. But with three or more candidates, things immediately get messy. These are too fairly straightforward to understand systems but they both have very undesirable properties about them. So as I was saying before we don't like to use the sort of non-specific language but there's no, with arrows theorem if you consider that to be a fair system then arrows theorem shows that there's no fair system and that's kind of a big I don't know maybe you think it's not a problem with democracy I don't know so used to decide. Okay, thank you. We've got about five minutes for questions and I can see I've got a question over there. So thanks very much. Something that I think is quite important is there's a difference between this voting system is massively and obviously flawed in everyday situations and it's possible to come up with a very complicated corner case where this system fails. So is there a way of quantifying that or quantifying how regularly these failure modes occur in real elections as opposed to in theory this system can produce this ridiculous result but it never actually does. That's a very good question, thank you. So some people, one estimate for if we applied alternative vote to UK elections was that it would hardly ever happen that the monotone criterion would fail that people would have been better voting differently. I think that that made quite specific assumptions about how voters would choose the parties in the future. It's really difficult to make inference about whether this ever does happen because people can't release the whole ranking of candidates. For example in Australia they use AV but there's no list of here's what everyone voted for because someone could say right, I want you to list there's 20 candidates list them in this order so that I know that you've done as I've said so we only have information about what was the first count what was the second count and so on. So it's really difficult to know whether this does happen with practical systems. First pass the post you also have majority loser thing. The thing is that most people know that's a thing. They know that if I vote for the candidate I really like I might be harming the candidate I do the worse the better of two evils so people sometimes do vote tactically based on that as well. They know that that's a problem and try to get round it. Does any of that change with multi-member systems where there's more than one winner? Sorry what was the first bit? Does it change with multi-member systems where there's more than one winner? So I assume it only applies to a single candidate. You can single winner election you still I think it I'm not 100% sure I think it still applies I'm not 100% sure on that. What's your opinion on concordants voting systems? What about them? I know they're computationally intensive and it's obviously very difficult for a lot of people to understand how they work and the more things that are involved the heavier the computation gets but is it possible to actually fix that with more technology and better algorithms? Yes so more complex algorithms for doing voting systems to exist the problem is that you have you would need to conduct the election on voting machines which I guess some people don't like and also people the advantage of something I first passed the post is that everyone understands how the votes accounted whereas with something that's much smarter system people don't really understand it as much I think if we were doing it again if we were starting the whole democracy again a smarter system that had fewer disadvantages but we need to educate everyone at the beginning about exactly why this was a better system sorry does that answer the question I'm not sure Is there any way to check whether a system would be liable to tactical voting? So there is a theorem I can't remember the name of it but it's kind of similar to kind of similar to Aero's theorem that says if everyone has knowledge about what other people are voting then every system is vulnerable to tactical voting Can we thank the speaker again?